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The following polynomials in two variables X 1 and X 2 are symmetric: + + + + (+) as is the following polynomial in three variables X 1, X 2, X 3: . There are many ways to make specific symmetric polynomials in any number of variables (see the various types below).
But the terms of P which contain only the variables X 1, ..., X n − 1 are precisely the terms that survive the operation of setting X n to 0, so their sum equals P(X 1, ..., X n − 1, 0), which is a symmetric polynomial in the variables X 1, ..., X n − 1 that we shall denote by P̃(X 1, ..., X n − 1). By the inductive hypothesis, this ...
Since the two sides are equal as polynomials, the coefficients of on both sides must be equal, and the result follows. Combinatorial proof We prove the recurrence relation using the definition of Stirling numbers in terms of permutations with a given number of cycles (or equivalently, orbits ).
the product of a 2-cycle and a 4-cycle such as (1 2 3 4)(5 6) maps to another such permutation such as (1 4 2 6)(3 5), accounting for the 90 remaining permutations. And the odd part is also conserved: a 2-cycle such as (1 2) maps to the product of three 2-cycles such as (1 2)(3 4)(5 6) and vice versa, there being 15 permutations each way;
Aside from polynomial functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric -tensors on a vector space is isomorphic to the space of homogeneous polynomials of degree on . Symmetric functions should not be confused with even and odd functions, which have a different sort of symmetry.
Formally, P is a symmetric polynomial if for any permutation σ of the subscripts 1, 2, ..., n, one has P(X σ(1), X σ(2), ..., X σ(n)) = P(X 1, X 2, ..., X n). Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by ...
An nth root of unity is a complex number whose nth power is 1, a root of the polynomial x n − 1. The set of all nth roots of unity forms a cyclic group of order n under multiplication. [1] The generators of this cyclic group are the nth primitive roots of unity; they are the roots of the nth cyclotomic polynomial.
The Newton identities also permit expressing the elementary symmetric polynomials in terms of the power sum symmetric polynomials, showing that any symmetric polynomial can also be expressed in the power sums. In fact the first n power sums also form an algebraic basis for the space of symmetric polynomials.